Substitutes Dominic Calvert-Lewin and Cenk Tosun were on target to sink a Palace side who had earlier missed a second-half penalty.
The Eagles almost took the lead when Cheikhou Kouyate headed against the underside of the bar on 37 minutes.
But their best chance came when they were awarded a penalty for a trip by Seamus Coleman on Wilfried Zaha.
However, Pickford’s outstretched leg kept out Luka Milivojevic’s centrally-placed spot-kick.
Five minutes after coming on, and with only four minutes remaining, Calvert-Lewin headed Everton in front from Ademola Lookman’s cross.
Tosun then raced on to Michael Keane’s clearance to finish off Palace.
Their third successive Premier League win takes Everton up to eighth place on 15 points. Place stay in 15th on seven points.
Everton: 28 Oct v Man Utd (A)
Crystal Palace: 28 Oct v Arsenal (H)